Skip-Gram Model Training
Almost, but no. When you maximize that objective function, you do so by adjusting the parameters $\phi$ and $\theta$. After you're done with training, you can use your word embeddings for other NLP tasks. You don't, however, do any prediction directly from the skip-gram model. 

To maximize the first term, co-occuring words must have large inner products. That is, they must be "similar". To maximize the second term**, the randomly sampled words must have a small inner product with $\phi_u$. That is, they must be "dissimilar" to $\phi_u$. Moving these word embeddings around in the vector space to make some words similar and others not is the only thing that happens during skip-gram training. 
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** $\sigma(-x)=1-\sigma(x)$, so maximizing $\sigma(-x)$ is minimizing $\sigma(x)$